This system uses the relationships of the prime octave (A3/A4) to the prime 5ths (A3/E4 & D4/A4) to determine the ideal widths for both the prime octave and the prime 5ths. This relationship will be found using the A4 Numbers, the 4th A4 Number, and Templates.
After the A4 numbers and the 4th A4 numbers are known, a template (stored in the Accu-Tuner’s memory) can be selected for Mapping A3.
The selected template will be used to tune A2, D3, E3, A3, D4, E4, and A4.
Start with the prime octave: tune A4, A3, D4, and E4 to the template.
The A4 Number information lets us know the width of the prime octave but the prime 5ths (A3/E4 & D4/A4) must be measured to learn their widths.
The prime 5ths:
After using the template to tune A4, A3, D4, and E3, most of the time the widths of the prime 5ths will be different. One may be -2.2 and the other -1.2. But at this point in the routine, all we need to know is the combined width of those two prime 5ths. We’ll balance them later (make them the same width).
Since a good starting point width for each of the prime 5ths is -1.5 c. each, what we are looking for at this point is a combined width of these two prime 5ths of -3.0 c. (-1.5 + -1.5 = -3.0)
If the combined width is > -3.0 (say -4.0) the prime octave is too narrow – by about 1.0 c. If the combined width is < -3.0 (say -2.0 c.) the octave is too wide and should be narrowed- by 1.0 c.
Whatever the amount is, either above or below the -3.0 target, the octave should be adjusted by that amount. Accurately adjusting the width of the prime octave is really easy using templates. To adjust the width of the prime octave, simply select a template with a slightly lower or higher A3 setting – by the amount needed and then re-tune A4,A3, D4 and E4.
Once those 4 notes have been re-tuned, remeasure the prime 5ths. Their combined widths should now be very close to -3.0 c.
Again, at this point, it doesn’t matter if they are not the same width. We will balance them later. But for now, all we’re looking for is a pair of prime 5ths whose combined widths equal -3.0 c.
On most pianos, at this point the prime octave should sound pretty good. When listening to the prime 5ths, and the resultant prime 4ths (A3/D4 & E4/A4) depending on how different the widths or the prime 5ths are, you may be able to hear a slight difference in the beating of the 5ths.
When the prime 5ths widths are different, most of the time, that unbalanced aspect of the tuning so far will show up in the resultant 4ths, since they are a slightly faster beating interval than the prime 5ths. So listening to the upper prime 4th, compared to the lower prime 4th will confirm not only the difference in the widths of the prime 5ths, but it will also indicate the direction in which the midpoint of the prime octave will need to adjusted to balance the prime 5ths, which will of course also balance the prime 4ths.
Once we know the difference between the widths of the prime 5ths, it is easy to determine how much each of them need to be moved so they are of equal width. If the lower 5th is -2.0 and the upper 5th is -1.0, lowering both by .5 will result in both of them being -1.5 and they will then be balanced.
To do this, simply look at the setting of D4 on the template and lower it by .5 c. and re-tune D4. Then do the same for E4. Once they’ve both been re-tuned to the new settings, both the prime 5ths and their resultant prime 4ths should sound pretty good.
And as long as the prime octave 2:1 isn’t wider than about 2.8 c., it should sound pretty good too.
We now know really good locations for A4, A3, D4 and E4 for this piano. These settings will be used for the rest of the mapping process and will also be used when creating the tuning using our Littau-Conrad Spreadsheet.
Again, this setup will work well on most pianos, but of course, not on every piano. But using these guidelines should give an idea as to how this works.
Being able to tune and measure accurately is very important, and it’s not easy. Accuracy is the key to the best sounding results. Practice may not make perfect, but eventually practice and experience will improve accuracy in both tuning and measuring.
The Prime Octave:
On most pianos as long as the width of the prime octave’s 2:1 is less than 3 c. wide, the beating will be tolerable for that octave.
Of course on some pianos, the width of the prime octave’s 2:1 will need to be greater than 3.0 c. But for now, and on most pianos, using that ‘no wider than 3.0 c.’ guideline is a good starting point.
The prime octave’s width can be determined by the widths of the prime 5ths. In a sense, it’s the 5ths that determine the width of the prime octave.
These widths, may be a bit subjective, since some can tolerate and actually enjoy a wider prime octave width. Letting the piano be the guide with the goal of minimal beating is always a good place to start.
I don’t want the prime octave to be any wider than necessary as it accommodates it’s prime 5ths. But the prime octave needs to be wide enough to accommodate a pair of good sounding prime 5ths.
Since it’s a piano, there are no hard and fast rules that work for every piano. Each piano is it’s own little laboratory so to speak.
But all the exceptions and solutions I’ve found are derivatives of this basic setup. Using this basic procedure lets me know very quickly if the piano in front of me will fall into the ‘most pianos’ category or not. But knowing the widths of the prime octave and prime 5ths during the process helps us to find the best sounding compromise since we are aware of the exact changes being made to both the octave and the 5ths.
There are pianos out there whose pure 4:2 (prime octave) results in a really wide and beating too fast 2:1. On those pianos the prime 4:2 actually ends up being narrow! But it’s not just a simple matter of contracting the octave because sometimes on those pianos, that makes the prime 5ths, too narrow and they start beating too fast.
So compromises are necessary on challenging pianos resulting in a prime octave’s 2:1 wider than 3.0 c, and the prime 5ths more narrow than -1.5 c.. These challenging pianos are a balancing act, trying to find the best sounding – or as Al Sanderson would say, the least bad sounding – compromise for both the octave and the 5ths.
I’ve tuned many of these challenging pianos and it took me a while to figure out what I needed to do to make them sound good. Using my standard guidelines was like trying to fit a square pet in a round hole. But I eventually figured it out. The guidelines above are starting points and will work on most pianos, but the more challenging pianos require more thoughtful and experienced treatment.
I think the biggest improvement we can make with them is to have a balanced pair of prime 5ths. Whatever that 5th width is, it must leave us with a tolerable sounding prime octave, and vice versa.
The octave may be beating more than we like, and so may the 5ths, but if those prime 5ths are ‘balanced’, the overall sound of the piano will be surprisingly satisfactory – the least bad sounding solution.
And not only are we then balancing the 5ths, but we are balancing the 5ths with the sound of the octave. We basically keep working back and forth until we find the least bad sounding 5ths that work with the least bad sounding octave.
I have developed a new appreciation for the designers who have had to work with such short string scales. I have also come to the not very ‘friendly’ conclusion that if they are properly tuned, they can be made to sound quite OK for what they are. But they (the challenging pianos) really require this type of treatment to sound as good as they can.
So getting back to the over all scheme of things, on most pianos, if the combined width of the 5ths is -3.0, and the prime octave’s 2:1 is less than 2.8 c or so wide, the Mapping of A3 is completed.
Balancing the Prime 5ths:
For this Prime Octave/Prime 5ths relationship to sound it’s best, the prime 5ths must be balanced. They must be the same width. Balancing the prime 5ths also balances the resultant prime 4ths (A3/D4 & E4/A4).
Balancing the prime 5ths involves raising or lowering the mid point of the prime octave. Even thought the mid point of the prime octave is D#4, the locations of D4 and E4 will be used to find D#4. Once we know where D4 and E4 should be, D#4 can easily be computer as the halfway point between D4 and E4.
For example if the settings for D4 is 5.0 and E4 is 6.4, then D#4 will be 5.7. Our spreadsheet displays the settings for D4, D#4 and E4 to facilitate the proper placement (raising or lowering) of the midpoint of the prime octave.
****Our spreadsheet also displays the settings for D3, D#3, and E3 in the 3rd octave so the 3rd octave’s mid point can be adjusted (raised or lowered) to it’s best sounding location. This mid point adjustment in the 3rd octave (the Sub-Prime octave) is not for balancing the sub prime 5ths. Of course it can be used for that, but it’s higher purpose is to adjust the shape of the curve within the sub prime octave to more accurately fit the scale of the piano in that area.
It’s in this 3rd octave where most of the issues occur going from long bridge to bass bridge, plain wire trichords to wound bichords, plain wire trichords to plain wire bichords, plain wire trichords to wound trichords, &c., so being able to adjust the mid point in this is really nice.
The same template used in the prime octave is used for mapping the sub prime octave. The sub prime octave is simply an extension of the relationship established in the prime octave.
Most of the time the sub-prime 5ths are not balanced as they were in the prime octave. Most of the time the upper 5th will be a little more narrow than the lower sub prime 5th, as the 5ths get slightly wider (less narrow) as the tuning extends down into the bass.
This mid point adjustment facilitates ending up with good sounding sub prime 5ths (A2/E3 & D3/A3), good sounding octaves (D3/D4, E3/E4, A2/A3), a good sounding double octave (A2/A4) and a pair of good sounding 12ths (A2/E4, D3/A4).
All of these settings are used by the LC spreadsheet to create the tuning which will pass through all of these target and mini target notes.***
Being able to adjust the mid point in the prime and sub prime octaves is a unique feature of this system.
If the lower prime 5th is -2.0 and the upper prime 5th is -1.0, the lower 5th is too narrow and the upper 5th is too wide. By simply raising the mid point of the prime octave (D#4) by .5 c., each of the prime 5ths will be -1.5 c.
The only way we can really appreciate this is to hear it for ourselves. That slight adjustment at the midpoint of the prime octave not only cleans up the 5ths, but it also cleans up the resultant 4ths, and of course all the other intervals within the prime octave – including the M3rds!
And being able to adjust the shape of the curve in the prime (and sub prime) and have it be a part of the tuning created by the LC spreadsheet, is really very unique.